Natural Log Rules lne 1 ln1 0 ln0 undefined ln 1 a lna ln a b lna lnb ln ab lna lnb Chapter 7 Integration Fundamental Theorem of Calculus Part 1 If F is an antiderivative of f F f Then f x dx F b F a Part 2 If f is continuous Then f t dt is an antiderivative of f b a x a 7 1 Integration by Substitution 1 Let w be the inside function 2 Reunite the integral as an integral in w 3 Evaluate the integral in w 4 convert w back to x Choices for u in order of preference Logarithm Inverse Trig Polynomial Exponential Trig 7 2 Integration by Parts udv uv vdu 7 3 Tables of Integrals Completing the Square Trig Functions sin2 x 1 2 cos2 x 1 2 1 cos2 x 1 cos2 x sin 2x 2 sinx cosx cos 2x cos2x sin2x 7 4 Algebraic Identities and Trigonometric Substitutions Partial Fractions Strategy for Integrating Rational Function P x Q x 1 If deg P deg Q use algebraic long division and use partial fractions on the remainder 2 If Q x is a product of distinct linear factors use partial fractions of the form 3 IF Q x has a repeated linear factor x c n use partial fractions of the form A x c A1 x c A 2 x c 2 An 1 x c n 4 If Q x contains an unfactorable quadratic factor q x try a partial fraction of the form Ax B q x Trigonometric Substitutions Sine Substitutions Use to simplify integrands involving a2 b2 Let x asin Tangent Substitutions Use to simplify integrands involving a2 x2 Let x atan 7 5 Approximating Definite Integrals Midpoint Rule Trapezoidal Rule TRAP n n n 2 Under or Over Estimate If f is increasing on a b then Error in Left n Right n 1 n 1 n2 Error in Mid n Trap n Error in Simp n 1 n4 If f is decreasing on a b then n f x dx b a If f is ccd on a b then TRAP n If f is ccu on a b then b a f x dx MID n MID n f x dx TRAP n b a Simpson s Rule 2MID n TRAP n 3 7 6 Improper Integrals Occur when trying to integrate functions whose graphs are infinite in extent either horizontally vertically or both Type 1 Horizontally Infinite Region Let f x be defined for x a If lim b b a f x dx exists and is finite f x dx converges a So a f x dx lim b f x dx b a Otherwise we say f x dx diverges a Type 2 Vertically Infinite Region vertical asymptotes Suppose f x is defined on a b and that f x or f x as x a If lim c a b c f x dx exists and is finite then f x dx converges b a and b a f x dx lim c a f x dx b c Otherwise f x dx diverges b a Type 3 Vertically and Horizontally infinite region basically a combination of types I and II which we handle by breaking up the interval of integration 7 7 Comparison of Improper Integrals The Comparison Test for Let f x 0 a f x dx 1 Guess by looking at the integrand for large x what the integrand behaves like and whether it converges or not This gives us a fn to compare the integrand with 2 Compare the guess by comparison If 0 f x g x and g x converges then f x dx converges If 0 g x f x and g x diverges then f x dx diverges a a a a Useful Integrals to Compare With 1 x p dx converges for p 1 diverges for p 1 1 1 1 x p dx converges for p 1 diverges for p 1 0 0 e ax dx converges for a 0 diverges for a 0 Limit Comparison Test Type I If f x g x are continuous and positive on 1 and lim x f x g x c exists with 0 c Then a f x dx and either both converge or both diverge g x a Chapter 8 Using the Definite Integral width w w h of horizontal cross section depends on height h Add all pieces together to get approx total area A 8 1 Areas and Volumes Areas by Horizontal Slices A w h d h Areas by Vertical Slices A h x dx b a b a Volumes by Horizontal Slices V b a A h dh Volumes by Vertical Slices V b a A x dx 8 2 Applications to Geometry To Compute a Volume Area Length Using an Integral Divide the solid region length into small pieces whose volume area length we can easily approximate Add the contributions of the pieces together gives us a Riemann Sum which approximates the total volume area length Take the limit as the number of terms in the sum tends to infinity Leads to a definite integral for the total volume area length Volumes of Revolution A x f x 2 V A x dx b a b a V f x 2 dx Arc Length S b 1 dx dy 2 a dx 8 3 Polar Coordinates cos x r x rcos y rsin y r sin Conversion of Polar Coordinates to Cartesian Coordinates Conversion of Cartesian Coordinates to Polar Coordinates y x tan y x tan x 0 x2 y2 r2 x 0 r2 x2 y2 So arctan y x and r x2 y2 not sufficient to tell in which quadrant should be Use angle measure to find correct quadrant What if x 0 arctan y x doesn t make sense But is y 0 then we need a value of which puts us on the positive y axis If y 0 then we need a value of which puts us on the negative y axis Finally if x y 0 and we re at the origin then r 0 and is undefined Except for this case given x y all possible values of differ by integer multiples of 2 like in the example above Curves in Polar Coordinates i r 2 This is just a circle of radius 2 about the origin ii 3 This is a half line with slope tan 3 3 which extends from the origin iii y 1 This is a horizontal line in Cartesian Coords iv r 2cos This is a circle of radius 1 centered at 1 0 v r 1 cos This curve is called a cardoid vi r 3sin 2 This curve is called a four leaved rose vii r2 a2cos 2 This curve is called a lemniscate Area in Polar Coordinates A 1 2 r2 d Slope in Polar Coordinates dy dx dy d dx d Arc Length in Polar Coordinates S dx d 2 dy d 2 d 8 4 Density and Center of Mass To find the total amount of some …